The escape of high-energy photons from gamma-ray bursts

Eleven bright gamma-ray bursts (GRBs) detected by BATSE have also been seen at much higher energies by EGRET, six at energies above 10 MeV. Most distinctive among these is GRB 940217, which includes long duration, hard gamma-ray emission, and the most energetic GRB photon detection to date, around 18 GeV. Such observations imply that these bursts are optically thin to photon-photon pair production at all observed energies, for target photons both internal and external to the source. For bursts more than about 30 pc away, internal transparency can be achieved only if the source is moving with a relativistic bulk Lorentz factor Gamma much greater than 1, or if the radiation is highly beamed. Early calculations of gamma gamma --> e(+)e(-) considerations for GRBs were limited to cases of a beam with opening half-angle Theta(B) similar to 1/Gamma, or expansions of infinitely thin spherical shells. This paper presents our extension of pair production optical depth calculations in relativistically expanding sources to more general geometries, including shells of finite thickness and arbitrary opening angle. The problem is reduced analytically to a single integral in the special, but quite broadly applicable, case of observing photons only along the axis of the expansion. We find that the minimum bulk Lorentz factor for the EGRET sources to be optically thin, i.e., display no spectral attenuation, is only moderately dependent on the shell thickness and virtually independent of its opening solid angle if Theta(B) greater than or similar to 1/Gamma. This insensitivity to Theta(B) relieves the commonly perceived number problem for nonrepeating sources at cosmological distances, i.e., it is not necessary to invoke small Theta(B) to effect photon escape. The values of Gamma obtained, typically of the order of 10 for halo bursts and greater than or similar to 100 for sources of cosmological origin, depend somewhat on the choice of GRB timescale used to determine the expansion size. Our new limits on required velocity for given source geometries will aid in placing realistic constraints on GRB source models.